Continuous dependence and differentiating solutions of a second order boundary value problem with average value condition

Lyons, Jeffrey; Major, Samantha; Seabrook, Kaitlyn

doi:10.2140/involve.2018.11.95

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Continuous dependence and differentiating solutions of a second order boundary value problem with average value condition

Jeffrey W. Lyons, Samantha A. Major and Kaitlyn B. Seabrook

Vol. 11 (2018), No. 1, 95–102

DOI: 10.2140/involve.2018.11.95

Abstract

Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem $y^{''} = f (x, y, y^{'})$ , $a < x < b$ , satisfying $y (x_{1}) = y_{1}$ , $1 ∕ (d - c) \int_{c}^{d} y (x) \underset{̣}{x} = y_{2}$ , where $a < x_{1} < c < d < b$ and $y_{1}, y_{2} \in ℝ$ with respect to each of the boundary data $x_{1}$ , $y_{1}$ , $y_{2}$ , $c$ , $d$ solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.

Keywords

continuous dependence, boundary data smoothness, average value condition, Peano's theorem

Mathematical Subject Classification 2010

Primary: 34B10

Milestones

Received: 3 August 2016

Revised: 13 August 2016

Accepted: 28 August 2016

Published: 17 July 2017

Communicated by Martin J. Bohner

Authors

Jeffrey W. Lyons
Department of Mathematics Nova Southeastern University 3301 College Ave Fort Lauderdale, FL 33314 United States

Samantha A. Major
Department of Mathematics Nova Southeastern University 3301 College Ave Fort Lauderdale, FL 33314 United States

Kaitlyn B. Seabrook
Department of Mathematics Nova Southeastern University 3301 College Ave Fort Lauderdale, FL 33314 United States

Vol. 11, No. 1, 2018